- producing the desired transient response
- reducing steady state errors
- achieving stability
![PID Controller Design Method PID Controller Design Method](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhf8-mjHz7h11Ashv-7-4Pddv5Bm51GYR5C6gjccEQW_8E7heAsvf6Kxjb_z1s7Z_27RHJ5X7EwOyNk2iKpBBv07pav4a78FYJGIBrUJHtpLQ7hH5gokbhuIqjCcg1jwleaNwL4Wp4JFLtB/w320-h141/1.png)
Plant: A system to be controlled.
Controller: Provides the excitation for the plant; Designed to control the overall system behavior.
There are several techniques available to the control systems engineer to design a suitable controller.
One of controller widely used is the proportional plus integral plus derivative (PID) controller, which has a transfer function:
![proportional plus integral plus derivative controller proportional plus integral plus derivative controller](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5Cgpuz9TugESiEHEHkR6mMkohXp-tZtYXO0Y3mKNEix0ZRZqLwiPT1IMu2njkvmtZLDt8C9Hp5mxW5lD-XrU7v7J0VwoLdqNZvMcnSvT25zOmFD6Y7uYLpyZeJybHHHd2jqWskIC073qk/w200-h66/1.png)
Kp = Proportional gain
KI = Integral gain
Kd = Derivative gain
The signal (u) just past the controller is now equal to the proportional gain (Kp) times the magnitude of the error plus the integral gain (Ki) times the integral of the error plus the derivative gain (Kd) times the derivative of the error.
![signal that past the controller signal that past the controller](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIm7RJinI-XIJH0lm-T47LCfY2A8yv6v_RuBtvHPHF_6DDuSNFBdPemnKn0wYqalTMQGhjxSyS-GIglfRcGMZcPiCdcef58RYKal5LcVEeEMhmMWA4e0NycAmN3CWmZzgT1mvdg5GRJ26e/w200-h55/1.png)
MATLAB Program:
Design requirements
Settling time less than 1 seconds
Overshoot less than 5%
Steady-state error less than 1%
Open-loop transfer function of the DC Motor:
Proportional Control:
j=3.2284e-6;
b=3.5077e-6;
k=0.0274;
r=4;
l=2.75e-6;
num=k;
kp =1.7;
den=[(j*l) ((j*r)+(l*b)) ((b*r)+k^2) 0];
sys=tf(num,den);
feedbk=feedback(kp*sys,1);
step(feedbk,0:.001:1)
Response Curve:
![Proportional Control Proportional Control](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjF632FWks8aNS1w4iGyX9HW9RHfWoj44u7CnjaQzyNbtidlmsMqGLkbZqNPV094y0qTZAsB_x9QNgTbs39EmpITIuLrucV0luDLPIZIDMYHKE6Ubl6-sU54fvKqSifCHT1KmtmiQwRam06/s400/Picture5.png)
Proportional+Integral Control:
J=0.01;
b=0.1;
K=0.01;
R=1;
L=0.5;
num=K;
den=[(J*L) ((J*R)+(L*b)) ((b*R)+K^2)];
open_sys= tf(num,den);
subplot(2,1,1),step (open_sys,0:0.1:3);
step (open_sys,0:0.01:3);
title('Step Response for the Open Loop System');
kp = 100;
ki = 200;
kd = 0 ;
controller=tf([kd kp ki],[1 0]);
sys=feedback(controller*open_sys,1);
subplot(2,1,2),step (sys,0:0.01:3);
title('Closed loop Step:ki=100 kp=200');
Response Curve:
![Proportional and Integral Control Proportional Integral Control](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9c7Viq4QBg_9k78CkHuv4cKvY48JgppdbsU9s56ewSCMsiUW1PypP9TFioY76boP4etcYj9Mi_98BhX7qd_NIt4LAtvBMfkM2DH3FlQ0zGI0uvKg_g_mgM5JWzjyUBHZ9CcXv9240DxfD/s400/Picture6.png)
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